Question 5.5: Determine the location of the centroid of the circular arc s...
Determine the location of the centroid of the circular arc shown.
STRATEGY: For a simple figure with circular geometry, you should use polar coordinates.
MODELING: The arc is symmetrical with respect to the x axis, so y = 0. Choose a differential element, as shown in Fig. 1.
ANALYSIS: Determine the length of the arc by integration.
L=\int{dL}=\int^{\alpha}_{-\alpha}r \ d\theta = r \int^{\alpha}_{-\alpha}{d\theta}=2r \alpha
The first moment of the arc with respect to the y axis is
\begin{matrix} Q_y&=& \int{x \ dL}=\int^{\alpha}_{-\alpha}{(r \ \cos \theta)(r \ d\theta)}=r^2\int^{\alpha}_{-\alpha}{\cos \theta d\theta} \\ &=&r^2[\sin \theta]^{\alpha}_{-\alpha} =2r^2\sin \alpha \end{matrix}
Since Q_y=\bar{x}L, you obtain
\bar{x}(2r\alpha)=2r^2 \sin \alpha \quad \quad \quad \quad \bar{x}=\frac{r \sin \alpha}{\alpha}
REFLECT and THINK: Observe that this result matches that given for this case in Fig. 5.8B.
